A far-from-CMC existence result for the constraint equations on manifolds with ends of cylindrical type

TL;DR

This paper demonstrates the existence of solutions to the Einstein constraint equations on manifolds with cylindrical ends, expanding the class of initial data sets available for such geometries in general relativity.

Contribution

It extends previous work by constructing solutions on manifolds with cylindrical ends for a broader class of metrics and second fundamental forms, without the constant mean curvature condition.

Findings

Existence of solutions for Yamabe positive, conformally asymptotically cylindrical metrics.

Construction of initial data sets with prescribed conformal class.

Solutions applicable to manifolds with ends of cylindrical type.

Abstract

We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chru\'sciel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given a Yamabe positive metric g, which is conformally asymptotically cylindrical on each end, and a 2-tensor K such that (tr K)^2 is bounded below away from zero and asymptotically constant, then we may find an initial data set (g',K') such that g' lies in the conformal class of g.